You are to consider two upgrade paths for a machine. The first possible path will replace the CPU by a new CPU that has clock rate three times the current CPU but otherwise the same architecture. The second possible path will replace the disk storage subsystem by a new one that has two times the effective throughput as the present system; neither disk subsytem will saturate the internal bus connecting the disk subsystem to the rest of the machine.

1.1 Assume that processes spend 0.55 fraction of time executing in the CPU without disk access, and 0.45 fraction of time using disk access. What is the speedup of each upgrade path under this scenario?

1.2 Assume that for the upgraded machine, the process mix will change; now the average process spends 0.25 fraction of time executing in the CPU without disk access, and 0.75 fraction of time using disk access. What is the speedup of each upgrade path under this other scenario?

1.3 Assume that the cost is $600 for the CPU upgrade and $700 for the disk subsystem upgrade. Under the two scenarios of (1.1) and (1.2) above, which upgrade would you recommend and why?

1.4 Let us redo the above with arbitrary values; that is, assume that the fraction of time executing in the CPU is fcpu and that only disk access is the other possible state (an oversimplification in the real world), with fraction of time fdisk = 1 - fcpu. Assume that one will change either the CPU or disk, and that the performance increase for the CPU is Scpu and for the disk is Sdisk. What is the speedup achieved by changing:

1.4.1 The CPU

1.4.2 The disk.

1.4.3 Assuming that the new CPU costs Ccpu and the disk costs Cdisk. As a function of fcpu, which is the best solution, change the disk or change the CPU? Note: your general answer in this case should provide the same result for the specific values used earlier in this question.
(Hint: to solve this question, you may need to consider the use of the methods of calculus, in which the best solution provides the most speedup for the total cost of the upgrade and in which the overall speedup depends not only upon the two S values but also upon the fraction the machine spends that in disk or CPU use.)